What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \ ? $$
Explanation: \begin{align*}
10^2\times N^2&=22^2\times55^2\\
&=\left(2^2\cdot11^2\right)\times\left(5^2\cdot11^2\right)\\
&=\left(2^2\cdot5^2\right)\times\left(11^2\cdot11^2\right)\\
&=10^2\times \left(11^2\right)^2\\
&=10^2\times 121^2
\end{align*} So $N=\boxed{121}$.